while( <DATA> ) {
    print if /^Header2:/ .. eof;
}

__DATA__
<lsmothers@example.com>
SMTP
0<001501c4db9b$db8b2680$2d01a8c0@ryand9v889t9uc>
.X-Intermail-Unknown-MIME-Type=unparsedmessage
Header2: <headertwo@example.com
Received: from server.cluster1.example.com ([10.20.201.160])
        line 12